Chaotic Sensors Boost Measurement Precision Even with Limited Access

Scientists are increasingly focused on improving the precision of measurements using quantum metrology, but current protocols often rely on complex entangled states and complete access to the sensing system. Harshita Sharma from the Birla Institute of Technology and Science Pilani, Sayan Choudhury from the Harish-Chandra Research Institute, and Jayendra N. Bandyopadhyay, also of the Birla Institute of Technology and Science Pilani, demonstrate a pathway to overcome these limitations by leveraging quantum chaotic sensors in a collaborative effort. Their research establishes that chaotic dynamics can enable Heisenberg-limited precision even with unentangled initial states and only partial access to the sensor, a significant advancement for practical quantum sensing. The team’s findings reveal optimal initial states for weakly chaotic systems and show that strong chaos provides robustness against initial state selection, achieving enhanced sensitivity with remarkably limited measurement access. This work highlights quantum chaos as a valuable and robust resource for quantum-enhanced sensing in realistic scenarios.

In the weakly chaotic regime, spin-coherent states placed at the edge of the regular islands in the mixed classical phase space serve as optimal initial states for enhanced sensitivity.

Conversely, in the strongly chaotic regime, Iα is insensitive to the choice of initial state, achieving quantum-enhanced sensitivity even when a low fraction (∼5%) of the qubits are accessible. Iα ∝N2t2. Several works have explored routes such as non-linearity and quantum criticality to reach this limit, though the preparation and preservation of these states in many-body systems remains experimentally challenging.

Partial access to the system can drastically decrease the QFI and, consequently, the sensing capabilities, motivating the need to devise mechanisms to enhance QFI that do not depend on careful preparation of probe states or global accessibility. Recent works demonstrate that information scrambling in collective spin systems can circumvent the state preparation challenge, and quantum chaotic systems can achieve increased sensitivity even with product states.

These protocols remain robust in the presence of noise, providing a striking example of resilient designs, and optimised measurement protocols address the partial accessibility challenge in quantum critical and light-matter coupled systems. This research presents an affirmative answer to the question of whether chaos can be harnessed for resilient quantum-enhanced sensing with partial access measurements, demonstrating that long-range interacting chaotic systems provide a route for resilient quantum-enhanced sensing even when a very small fraction of the system is accessible.

The sensor is modelled as the quantum kicked top (QKT), where the strength of the chaoticity is controlled by tuning the non-linearity, serving as a powerful resource for quantum sensing when globally accessible measurements are performed. Strikingly, chaotic dynamics enables a large QFI even when a small fraction (∼10%) of the system is accessible. For a moderately chaotic case, the QFI under partial accessibility exhibits a strong dependence on the initial state, with coherent states localized at the edges of regular islands exhibiting a larger QFI at late times and achieving Heisenberg scaling in time.

In the strongly chaotic regime, there is no initial state dependence, and the temporal growth of Iα in the partially accessible QKT mirrors that of the fully accessible QKT. Consequently, quantum chaotic sensors retain their resilience in the presence of partial accessibility, providing a robust platform for quantum metrology. U = e−i κ 2j J2y e−iαJz Ja i = ∑i σa i are the collective angular momentum operators satisfying [Ji,Jk] = ιεiklJl and j = N/2.

They investigate the behaviour of the QFI for the QKT sensor composed of N = 1000 qubits (j = 500), both in the moderately chaotic (κ = 3) and the strongly chaotic (κ = 30) regime, analysing the time-evolution of the system prepared in unentangled SU spin-coherent states: |θ,φ⟩= eiθ(Jx sinφ−Jy cosφ) |j, j⟩. In the moderately chaotic regime, the QFI exhibits a strong initial state dependence, illustrated by examining the dynamics for three initial states: a non-equatorial island state |2.20,2.44⟩, chaotic sea state |2.46,0.32⟩, and edge state |2.56,2.31⟩.

At early times, the chaotic state exhibits the largest QFI value, but is overtaken by the edge state at later times. As the number of accessible qubits increases, fluctuations in the QFI are suppressed, and the metrological advantage of chaotic and edge states over regular island states persists even under partial access. For subsystem sizes up to 50% accessibility, the QFI exhibits robust scaling with time, approximately resembling the fully accessible case.

The Ehrenfest time tE ≃ln(2J)/λE and the Heisenberg time tH ≃J/3 are characteristic time scales for the quantum dynamics of the QKT. The QFI nearly reaches the same order as the full accessible case with as little as 5% access at later times, and the dependence of QFI with different subsystem sizes Q is shown for three representative times, varying Q from 1 to 900 qubits.

A central component of this work is the detailed analysis of quantum Fisher information (QFI) to quantify sensitivity in many-body sensors, focusing on classically chaotic systems and how initial states propagate within their mixed classical phase space. Initial states were deliberately chosen to be unentangled, circumventing the typical requirement for highly entangled probes in quantum metrology, motivated by the desire to demonstrate enhanced sensing capabilities without complex entanglement.

The study employed numerical simulations to track the evolution of the QFI over time, allowing for precise determination of the scaling behaviour with system size, performed on systems exhibiting both weak and strong chaotic regimes. Coherent states, positioned strategically at the edges of regular islands within the phase space, were identified as optimal initial states in the weakly chaotic regime, while the strongly chaotic regime demonstrated insensitivity to the initial state selection.

A key methodological innovation was the investigation of sensitivity under conditions of partial measurement accessibility, modelling scenarios where only approximately 0.222 of the system’s degrees of freedom are directly measurable. This realistic constraint was incorporated to assess the robustness of quantum-enhanced sensing in practical experimental settings.

Initial QFI values demonstrate Heisenberg scaling with time in quantum chaotic sensors, even when utilising unentangled initial states, with Iα increasing proportionally to the square of time. In the weakly chaotic regime, spin-coherent states positioned at the boundaries of regular islands within the classical phase space prove to be optimal initial states for maximising sensitivity.

Conversely, in the strongly chaotic regime, the QFI becomes independent of the initial state chosen, maintaining quantum-enhanced sensitivity even with limited accessibility when approximately 5% of the qubits are accessible. The phase space averaged Lyapunov exponent, a measure of chaoticity, accurately captures the system’s sensitive dependence on initial conditions, calculated by averaging over 40,000 trajectories evolved over 50,000 drive periods.

Phase space maps generated for different values of the non-linearity parameter, κ, visually confirm the transition from regular to chaotic dynamics, with distinct regular islands visible for κ = 3.0 and strong chaoticity with no discernible islands for κ = 30.0. The study demonstrates that even with partial access measurements, quantum chaotic systems provide a resilient platform for quantum metrology.

Scientists have long recognised the potential of quantum sensors to outperform their classical counterparts, but realising this promise demands exquisitely controlled, highly entangled states and precise measurement. This work sidesteps a significant hurdle by demonstrating that robust quantum enhancement is possible even when full control and measurement access are absent.

The research reveals that quantum chaos can create the necessary sensitivity, even starting with simple, unentangled initial states, departing from conventional wisdom that prioritises complex entanglement. This finding that chaotic systems can effectively ‘self-entangle’ during the sensing process opens up new avenues for designing practical quantum sensors, potentially leading to devices less reliant on maintaining fragile quantum states and more resilient to imperfections. Potential applications span diverse fields, from medical imaging and materials science to gravitational wave detection and navigation, with benefits most pronounced in strongly chaotic regimes.